By [1]}, if \(f\in W_0^{1,\Phi }(G)\) , then \(\bar{f}\in W^{1,1}(\mathbb {R}^n)\) . Since Sobolev space \(W^{1,1}(\mathbb {R}^n)\)  is the subspace of \(BV(\mathbb {R}^n)\)  (see e.g. [2]}), \(\bar{f}\in BV(\mathbb {R}^n)\) . By [2]}, if \(f\in BV(\mathbb {R}^n)\) , then the superlevel set \([f]_h\)  is of finite perimeter for \(\mathcal {L}^1\) -a.e. \(h>0\) .
